Variation Functions | Moosa Academy

Variation Functions — Moosa Academy

Some quantities rise together, some fall as the other rises, and some depend on two things at once. Variation gives each of these patterns an equation built around a single fixed number — the constant of variation — which you can recover from one pair of matching values and then reuse for every other pair.

Concept The three kinds of variation

Direct —

y

grows as

x

grows

Inverse —

y

shrinks as

x

grows

Joint —

y

varies with

x

and

z

together

In every case the number $k$ is the constant of variation, and $k \neq 0$ .

Type	Equation	Proportion	Graph
Direct	$y = kx$	$\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$	Straight line through the origin
Inverse	$y = \dfrac{k}{x}$	$x_1 y_1 = x_2 y_2$	Hyperbola
Joint	$y = kxz$	$\dfrac{y_1}{x_1 z_1} = \dfrac{y_2}{x_2 z_2}$	A flat surface

Theorem Why the proportion works

Because $k$ is the same for every pair of values, two pairs from the same relationship can be set equal to each other — and $k$ never has to be calculated.

Direct variation gives

k = \dfrac{y}{x}

, so

\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}

Inverse variation gives

k = xy

, so

x_1 y_1 = x_2 y_2

Notice the inverse proportion multiplies where the direct one divides. That single difference is what turns a rising line into a falling curve.

Example Direct variation

$y$ varies directly with $x$ , and $y = 15$ when $x = 5$ . Find $y$ when $x = 7$ .

Set up the proportion:

\dfrac{15}{5} = \dfrac{y}{7}

Cross multiply:

5y = 15 \times 7 = 105

Divide by

5

y = 21

$y = 21$ when $x = 7$

Here $k = 15 \div 5 = 3$ , so the relationship is $y = 3x$ . Substituting $x = 7$ directly gives the same $21$ .

Example Joint variation

$y$ varies jointly with $x$ and $z$ , and $y = 20$ when $x = 5$ and $z = 3$ . Find $y$ when $x = 9$ and $z = 2$ .

Set up the proportion:

\dfrac{20}{5 \times 3} = \dfrac{y}{9 \times 2}

Simplify the products:

\dfrac{20}{15} = \dfrac{y}{18}

Cross multiply:

15y = 20 \times 18 = 360

Divide by

15

y = 24

$y = 24$ when $x = 9$ and $z = 2$

Joint variation is direct variation with two partners instead of one — the product $xz$ simply takes the place of $x$ .

Example Inverse variation

$a$ varies inversely with $b$ , and $a = 28$ when $b = 2$ . Find $a$ when $b = 10$ .

Set up the proportion:

28 \times 2 = a \times 10

Simplify:

56 = 10a

Divide by

10

a = 5.6

$a = 5\dfrac{3}{5}$ when $b = 10$

The value of $b$ grew five times larger, so $a$ became five times smaller. That trade-off is the signature of inverse variation, and here $k = 56$ stays fixed throughout.

Summary

Direct variation is $y = kx$ — a straight line through the origin.
Inverse variation is $y = \dfrac{k}{x}$ — a hyperbola, falling as $x$ rises.
Joint variation is $y = kxz$ , direct variation with two partners.
The constant $k$ is fixed for the whole relationship and can never be zero.
Direct problems divide across the proportion; inverse problems multiply.
One matching pair of values is enough to answer any other pair.